Mathematics, as the authors say, like other familiar things, is easier to describe and to illustrate than to define: it is hard to find one general idea that will embrace all the manifold applications. But the impression we have received after reading this book is that mathematics is an attempt to systematize the universe: first some small part of the universe, then a larger part, and so on, until we begin to suspect that we shall need a formula as complex as the universe itself if we are to embrace all. As viewed from the standpoint of the intellect, mathematics is the most abstract and fundamental of all the sciences, being a sort of master-science underlying all the rest. To give the layman some notion of what it is all about has been the laudable aim of these writers, who have produced a most interesting and informative book.
First we have chapters on numbers, especially very large numbers. It is explained that primitive minds count without the use of numbers, by the simple process of matching one plurality with another; and that the idea of number apart from a particular plurality of objects is an abstraction which comes later. The existence of the very large numbers is an inference rather than a conclusion from observation. One very large number, facetiously called a "googol," is 10 to the power of 100, or a 1 with a hundred zeros after it. But Eddington estimates that the number of electrons in the entire universe is only 10 to the power of 79. If we wish to climb to farther heights we can speak of 10 to the power of one googol, which would be represented by a 1 with so many zeros after it that there would not be room to write them if you wrote one every inch all the way to the farthest nebula. Such a number may have no use in mensuration, but it may in the calculus of probabilities, as estimating the probability that my typewriter may suddenly jump off the table and land on the bed, or that an electron may do some excessively unusual thing.
Then follow chapters on the meanings of the word "infinite," the infinitely large, the infinitely small, the unbounded, the undivided; and the numerous paradoxes and fallacies which arise from a neglect to distinguish these meanings. The very big and the infinite are entirely different; there is no point where the very big begins to merge into the infinite; however large a number or magnitude you may reach, you will still be no nearer the infinite than you were at the start. The idea of the infinite is arrived at through "reasoning by recurrence': what we have done once we can do again, and we assume that we can go on doing it indefinitely. The paradoxes of Zeno are considered. Some of them are due to his failure to recognise that an infinite series may have a finite sum, and some depend upon a relation between position and motion which has engaged the attention of the profoundest mathematical minds. Leibniz tried to explain the infinitely small to Queen Sophie Charlotte of Prussia, but she said the behavior of her courtiers had made her so familiar with it that she did not need to be taught.
The principle of logarithms is clearly explained, with the remark that it is wonderful it was not thought of before. This may suggest to some that it has been thought of before; many of our mathematical discoveries may be rediscoveries, and there may have been things known to our remote ancestors that have not yet been rediscovered. Transcendental and imaginary numbers, π i and e, come in for consideration, and a great many formulas for approximations to π are given.
In the sections on assorted geometries we are introduced to the four-dimensionalists and flatlanders, and it is shown how they neglect to distinguish between physics and pure geometry, between physical space (that of our sensory experiences) and ideal space. Euclidean space is ideal and abstract; a three-dimensional geometric solid is as imaginary as a two-dimensional figure, and quite distinct from a solid physical body. The word "dimension," as used in mathematics, does not necessarily mean an extended line at right angles to other lines. Its meaning must be abstracted from that of physical space, and it is better to substitute the word "co-ordinate." Physical space is a three-dimensional manifold. When mathematicians speak of multi-dimensional manifolds they have no reference to physical space; such manifolds (or "spaces') have only the properties expressed by the postulates and axioms of the particular geometry concerned.
Euclidean geometry is shown to be only a theory, in which the propositions are deduced with faultless logic from the axioms and postulates laid down. But Lobachevsky, Bolyai, and Riemann have constructed equally consistent geometries upon other postulates, and these geometries are better adapted for the mensuration of curved surfaces than is Euclid's. Here again we see that geometries are an indefinite number of different systems invented for the purpose of co-ordinating certain groups of experiences which we have of the universe.
The chapter on mathematical puzzles will prove of great interest to many readers; we have here quite a large selection of such posers: various versions of the problem of a man crossing by a ferry and having with him a fox, a goose, and a bag of corn, only one of which he can carry at a time; the problem of the eight-quart jug of wine which is to be divided into two equal shares by means of jugs holding five and three quarts; mechanical puzzles with wires and rings; etc. The International Beer-Drinking Problem is as follows:
In a town on the border between the United States and Mexico the state of exchange was such that an American dollar was worth only 90 cents in Mexico, while a Mexican dollar was worth only 90 cents in America. So a man crosses the border into Mexico and orders ten cents worth of beer, which he pays for with a Mexican dollar, receiving an American dollar as change. With this he recrosses the border and buys another ten cents worth of beer with his American dollar, receiving a Mexican dollar as change. He keeps this up all day until he is quite full of beer. Question — who pays for the beer?
Another chapter is on paradoxes or inconsistencies, many of them hoary with antiquity. The paradoxes of Zeno naturally come in for consideration. That of Achilles and the Tortoise depends on the fact that Zeno confuses an infinite distance with a finite distance infinitely subdivided; but his problem of the moving arrow involves considerations which have exercised the greatest mathematical minds. Motion, when analysed by our logical mind, appears to become reduced to a series of disconnected stationary positions, which our perceptions synthesize in a way similar to that by which the separate images on the film are made to resemble figures in motion. In the same way a continuous line may be analysed into a series of separate points, or a continuous solid body into an aggregate of disconnected atoms. It would seem that the analytical function of the mind is bound to lead to dilemmas; and is not this what Zeno was aiming to demonstrate?
The section on chance and probability carries us into deep water. The calculus of probabilities succeeds to perfection on the large scale, but breaks down utterly in the details. It is thanks to the former virtue that our insurance companies can profitably enable us to sit in our armchair and smoke at forty for the rest of our life; but they cannot for the life of them tell whether it is going to rain tomorrow. I can toss a coin a thousand times with the assurance that the proportion of heads and tails will be almost exactly 50 — 50; but I have not the least idea, if I only make one toss, whether it will be heads or tails. According to the rules the chance of heads or tails is always equal, no matter how often I may have thrown heads in succession. But what about the betting odds?
Edgar Poe calls attention to this discrepancy: the fact of sixes having been thrown twice with dice is warrant for betting the largest odds that sixes will not be thrown a third time; yet the intellect cannot see how the past throws can influence the ones that follow them. The error involved, he says, is one of an infinite series of mistakes which arise in the path of Reason through her propensity for seeking truth in detail. This can only mean that the successive throws are connected; that the Reason is wrong and the intuition right.
The kinetic theory of gases explains their expansive pressure by an elaborate integration of the kinetic energies of all the molecules; but when we consider a single molecule, we visualize it as flying about at its own free will like a midge in a sunbeam. This may help us to reconcile our doubts about the compatibility of free will with law. If you only have a sufficient number of individual free wills, they will together fulfil the law: the calculus of probability proves it.
The authors point out that classical physics regarded as immutable laws of Nature phenomena which had been shown to hold good within a limited range of experience; and that modern physics has extended its observations to regions wherein some of those rules do not hold good. Thus natural law has been largely replaced, perhaps even ousted, by statistical inference. The laws of Nature are no longer regarded as "simple and constant." We used to say: "All men die; Socrates is a man; therefore Socrates will die." Now we say: "All men die so far as we know; Socrates is a man; therefore he will probably die."
Mathematics is formulation, and formulation is a method of approximating the truth by stages. Such is the impression we receive.
A word should be said on the subject of abstractions. The meaning of this word depends on the point of view. If moving bodies are considered as real, then motion becomes an abstraction from this reality. But if motion itself be regarded as real, then moving bodies must be regarded as only particular manifestations of motion. Is the infinite merely the negation of the finite? Or is the finite a limitation of the infinite? Which is the prior term? Descartes held that the infinite is the positive idea, and the finite the negative, and that therefore the infinite presupposes the finite. Kant held that space is not a general conception, abstracted from particular ideas of space, but that particular spaces are arrived at by limitations of the one infinite space that is prior to them. This is of course the way space is viewed in The Secret Doctrine; and so also is number. Number, which from our viewpoint is an abstraction, is viewed as a reality. That may seem unthinkable, but the universe is not likely to be conditioned by the infirmities of our conceptual powers. We can form no idea of what a number is in itself; we can only think of a plurality of objects or parts. So long as we have to rely on the argumentative and analysing faculties, we are bound to come up against dualities, antinomies, contradictions; and mathematics has sought to palliate the difficulty by all sorts of subtle and ingenious devices. But Theosophists believe that the human mind is able to rid itself of this kind of Maya and thus to be able to grasp ideas beyond the scope of our ordinary powers. The book concludes with a chapter on the Calculus and the meaning of Change and Changeability as understood by mathematicians. Much praise must be given to the excellent illustrations and diagrams by Rufus Isaacs.
1. Mathematics and the Imagination. By Edward Kasner and James Newman. Simon and Schuster: New York: 1940. $2.75. (return to text)